Positive real numbers question

[TTob]

question:
let x_1,...,x_n positive real numbers.
prove that

\lim_{p\to \infty}\left(\frac{x_1^p+...+x_n^p}{n}\right)^{1/p}=max\{x_1,...,x_n\}

can you give me some hints ?

 

[Dick]

Suppose the largest number is x_k. What is the limit of (x_i/x_k)^p? Is that enough of a hint?

 

[TTob]

this limit equals 1 when x_i=x_k and 0 when x_i<x_k.
so lim (x_1^p+...+x_n^p)/x_k^p = number of x_i that equal to x_k.
I don't know how to continue.

 

[Dick]

Factor x_k^p out of the sum in your limit (that's how you get that expression whose limit you just figured out). Then bring it outside of the ()^(1/p) power as x_k.

 

[TTob]

Thank you !